3.47 \(\int \frac {\text {csch}^3(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\)

Optimal. Leaf size=213 \[ -\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a d (a+b)^3 \left (a \cosh ^2(c+d x)+b\right )}-\frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{3/2} d (a+b)^4}+\frac {b (2 a-b) \cosh (c+d x)}{4 a d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 d (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}+\frac {(a-5 b) \tanh ^{-1}(\cosh (c+d x))}{2 d (a+b)^4} \]

[Out]

1/2*(a-5*b)*arctanh(cosh(d*x+c))/d/(a+b)^4+1/4*(2*a-b)*b*cosh(d*x+c)/a/(a+b)^2/d/(b+a*cosh(d*x+c)^2)^2-1/8*(4*
a^2-9*a*b-b^2)*cosh(d*x+c)/a/(a+b)^3/d/(b+a*cosh(d*x+c)^2)-1/2*cosh(d*x+c)*coth(d*x+c)^2/(a+b)/d/(b+a*cosh(d*x
+c)^2)^2-1/8*(15*a^2-10*a*b-b^2)*arctan(cosh(d*x+c)*a^(1/2)/b^(1/2))*b^(1/2)/a^(3/2)/(a+b)^4/d

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Rubi [A]  time = 0.34, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4133, 470, 578, 527, 522, 206, 205} \[ -\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a d (a+b)^3 \left (a \cosh ^2(c+d x)+b\right )}-\frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{3/2} d (a+b)^4}+\frac {b (2 a-b) \cosh (c+d x)}{4 a d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 d (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}+\frac {(a-5 b) \tanh ^{-1}(\cosh (c+d x))}{2 d (a+b)^4} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

-(Sqrt[b]*(15*a^2 - 10*a*b - b^2)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(8*a^(3/2)*(a + b)^4*d) + ((a - 5*b
)*ArcTanh[Cosh[c + d*x]])/(2*(a + b)^4*d) + ((2*a - b)*b*Cosh[c + d*x])/(4*a*(a + b)^2*d*(b + a*Cosh[c + d*x]^
2)^2) - ((4*a^2 - 9*a*b - b^2)*Cosh[c + d*x])/(8*a*(a + b)^3*d*(b + a*Cosh[c + d*x]^2)) - (Cosh[c + d*x]*Coth[
c + d*x]^2)/(2*(a + b)*d*(b + a*Cosh[c + d*x]^2)^2)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 578

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(g^(n - 1)*(b*e - a*f)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c -
a*d)*(p + 1)), x] - Dist[g^n/(b*n*(b*c - a*d)*(p + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*S
imp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a*f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x], x] /; Fre
eQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]

Rule 4133

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = F
reeFactors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[((1 - ff^2*x^2)^((m - 1)/2)*(b + a*(ff*x)^n)^p)/(ff*x)^(n*
p), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p
]

Rubi steps

\begin {align*} \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2 \left (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 b+(-a+2 b) x^2\right )}{\left (1-x^2\right ) \left (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{2 (a+b) d}\\ &=\frac {(2 a-b) b \cosh (c+d x)}{4 a (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {2 (2 a-b) b-2 \left (2 a^2-8 a b-b^2\right ) x^2}{\left (1-x^2\right ) \left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{8 a (a+b)^2 d}\\ &=\frac {(2 a-b) b \cosh (c+d x)}{4 a (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a (a+b)^3 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {-2 (11 a-b) b^2+2 b \left (4 a^2-9 a b-b^2\right ) x^2}{\left (1-x^2\right ) \left (b+a x^2\right )} \, dx,x,\cosh (c+d x)\right )}{16 a b (a+b)^3 d}\\ &=\frac {(2 a-b) b \cosh (c+d x)}{4 a (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a (a+b)^3 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {(a-5 b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 (a+b)^4 d}-\frac {\left (b \left (15 a^2-10 a b-b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{8 a (a+b)^4 d}\\ &=-\frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{3/2} (a+b)^4 d}+\frac {(a-5 b) \tanh ^{-1}(\cosh (c+d x))}{2 (a+b)^4 d}+\frac {(2 a-b) b \cosh (c+d x)}{4 a (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a (a+b)^3 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}\\ \end {align*}

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Mathematica [C]  time = 4.12, size = 524, normalized size = 2.46 \[ \frac {\text {sech}^5(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\frac {\sqrt {b} \left (-15 a^2+10 a b+b^2\right ) \text {sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left (\frac {\sinh (c) \tanh \left (\frac {d x}{2}\right ) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )}{a^{3/2}}+\frac {\sqrt {b} \left (-15 a^2+10 a b+b^2\right ) \text {sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left (\frac {\sinh (c) \tanh \left (\frac {d x}{2}\right ) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )}{a^{3/2}}-\frac {8 b^2 (a+b)^2}{a}+\frac {2 b (9 a+b) (a+b) (a \cosh (2 (c+d x))+a+2 b)}{a}-(a+b) \text {sech}^2\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2+4 (a-5 b) \text {sech}(c+d x) \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right ) (a \cosh (2 (c+d x))+a+2 b)^2-(a+b) \text {csch}^2\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2-4 (a-5 b) \text {sech}(c+d x) \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right ) (a \cosh (2 (c+d x))+a+2 b)^2\right )}{64 d (a+b)^4 \left (a+b \text {sech}^2(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^5*((-8*b^2*(a + b)^2)/a + (2*b*(a + b)*(9*a + b)*(a + 2*b + a*C
osh[2*(c + d*x)]))/a + (Sqrt[b]*(-15*a^2 + 10*a*b + b^2)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh
[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/
Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x])/a^(3/2) + (Sqrt[b]*(-15*a^2 + 10*a*b + b^2)*ArcTan[(
(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]
*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x])/a^(3/2)
- (a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Csch[(c + d*x)/2]^2*Sech[c + d*x] + 4*(a - 5*b)*(a + 2*b + a*Cosh
[2*(c + d*x)])^2*Log[Cosh[(c + d*x)/2]]*Sech[c + d*x] - 4*(a - 5*b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Log[Sinh
[(c + d*x)/2]]*Sech[c + d*x] - (a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[(c + d*x)/2]^2*Sech[c + d*x]))/(
64*(a + b)^4*d*(a + b*Sech[c + d*x]^2)^3)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[84,-86]Warning, need to choose a branch for the root of a polynomial with parameters. Thi
s might be wrong.The choice was done assuming [a,b]=[-42,-12]Warning, need to choose a branch for the root of
a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-43,-99]Warning, need to
choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assumin
g [a,b]=[-28,94]Warning, need to choose a branch for the root of a polynomial with parameters. This might be w
rong.The choice was done assuming [a,b]=[-7,46]Warning, need to choose a branch for the root of a polynomial w
ith parameters. This might be wrong.The choice was done assuming [a,b]=[-35,-99]Warning, need to choose a bran
ch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[7,50]
Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice
was done assuming [a,b]=[-63,-70]Undef/Unsigned Inf encountered in limitEvaluation time: 1.54Limit: Max order
reached or unable to make series expansion Error: Bad Argument Value

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maple [B]  time = 0.45, size = 1555, normalized size = 7.30 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^3/(a+b*sech(d*x+c)^2)^3,x)

[Out]

1/8/d*tanh(1/2*d*x+1/2*c)^2/(a^3+3*a^2*b+3*a*b^2+b^3)+9/4/d*b/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+
1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*a^2*tanh(1/2*d*x+1/2*c)^6-5/4/d*b^2/(a+b)^
4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*
a*tanh(1/2*d*x+1/2*c)^6-13/4/d*b^3/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2
*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^6+1/4/d*b^4/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*ta
nh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/a*tanh(1/2*d*x+1/2*c)^6+27/4/d*
b/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b
+a+b)^2*a^2*tanh(1/2*d*x+1/2*c)^4-21/4/d*b^2/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1
/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*a*tanh(1/2*d*x+1/2*c)^4+29/4/d*b^3/(a+b)^4/(tanh(1/2*d*x+1/
2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c
)^4-3/4/d*b^4/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*
x+1/2*c)^2*b+a+b)^2/a*tanh(1/2*d*x+1/2*c)^4+27/4/d*b/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+
2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*a^2*tanh(1/2*d*x+1/2*c)^2+1/4/d*b^2/(a+b)^4/(tanh(1
/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*a*tanh(1/
2*d*x+1/2*c)^2-23/4/d*b^3/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2
*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^2+3/4/d*b^4/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*
x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/a*tanh(1/2*d*x+1/2*c)^2+9/4/d*b/(a+b)^4/
(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*a^
2+17/4/d*b^2/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x
+1/2*c)^2*b+a+b)^2*a+7/4/d*b^3/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^
2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2-1/4/d*b^4/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh
(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/a-15/8/d*b/(a+b)^4*a/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh
(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2))+5/4/d*b^2/(a+b)^4/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^
2+2*a-2*b)/(a*b)^(1/2))+1/8/d*b^3/(a+b)^4/a/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*
b)^(1/2))-1/8/d/(a+b)^3/tanh(1/2*d*x+1/2*c)^2-1/2/d/(a+b)^4*ln(tanh(1/2*d*x+1/2*c))*a+5/2/d/(a+b)^4*ln(tanh(1/
2*d*x+1/2*c))*b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/2*(a - 5*b)*log((e^(d*x + c) + 1)*e^(-c))/(a^4*d + 4*a^3*b*d + 6*a^2*b^2*d + 4*a*b^3*d + b^4*d) - 1/2*(a - 5
*b)*log((e^(d*x + c) - 1)*e^(-c))/(a^4*d + 4*a^3*b*d + 6*a^2*b^2*d + 4*a*b^3*d + b^4*d) - 1/4*((4*a^3*e^(11*c)
- 9*a^2*b*e^(11*c) - a*b^2*e^(11*c))*e^(11*d*x) + (20*a^3*e^(9*c) + 23*a^2*b*e^(9*c) - 29*a*b^2*e^(9*c) + 4*b
^3*e^(9*c))*e^(9*d*x) + 2*(20*a^3*e^(7*c) + 57*a^2*b*e^(7*c) + 47*a*b^2*e^(7*c) - 2*b^3*e^(7*c))*e^(7*d*x) + 2
*(20*a^3*e^(5*c) + 57*a^2*b*e^(5*c) + 47*a*b^2*e^(5*c) - 2*b^3*e^(5*c))*e^(5*d*x) + (20*a^3*e^(3*c) + 23*a^2*b
*e^(3*c) - 29*a*b^2*e^(3*c) + 4*b^3*e^(3*c))*e^(3*d*x) + (4*a^3*e^c - 9*a^2*b*e^c - a*b^2*e^c)*e^(d*x))/(a^6*d
+ 3*a^5*b*d + 3*a^4*b^2*d + a^3*b^3*d + (a^6*d*e^(12*c) + 3*a^5*b*d*e^(12*c) + 3*a^4*b^2*d*e^(12*c) + a^3*b^3
*d*e^(12*c))*e^(12*d*x) + 2*(a^6*d*e^(10*c) + 7*a^5*b*d*e^(10*c) + 15*a^4*b^2*d*e^(10*c) + 13*a^3*b^3*d*e^(10*
c) + 4*a^2*b^4*d*e^(10*c))*e^(10*d*x) - (a^6*d*e^(8*c) + 3*a^5*b*d*e^(8*c) - 13*a^4*b^2*d*e^(8*c) - 47*a^3*b^3
*d*e^(8*c) - 48*a^2*b^4*d*e^(8*c) - 16*a*b^5*d*e^(8*c))*e^(8*d*x) - 4*(a^6*d*e^(6*c) + 7*a^5*b*d*e^(6*c) + 23*
a^4*b^2*d*e^(6*c) + 37*a^3*b^3*d*e^(6*c) + 28*a^2*b^4*d*e^(6*c) + 8*a*b^5*d*e^(6*c))*e^(6*d*x) - (a^6*d*e^(4*c
) + 3*a^5*b*d*e^(4*c) - 13*a^4*b^2*d*e^(4*c) - 47*a^3*b^3*d*e^(4*c) - 48*a^2*b^4*d*e^(4*c) - 16*a*b^5*d*e^(4*c
))*e^(4*d*x) + 2*(a^6*d*e^(2*c) + 7*a^5*b*d*e^(2*c) + 15*a^4*b^2*d*e^(2*c) + 13*a^3*b^3*d*e^(2*c) + 4*a^2*b^4*
d*e^(2*c))*e^(2*d*x)) - 8*integrate(1/32*((15*a^2*b*e^(3*c) - 10*a*b^2*e^(3*c) - b^3*e^(3*c))*e^(3*d*x) - (15*
a^2*b*e^c - 10*a*b^2*e^c - b^3*e^c)*e^(d*x))/(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4 + (a^6*e^(4*c) +
4*a^5*b*e^(4*c) + 6*a^4*b^2*e^(4*c) + 4*a^3*b^3*e^(4*c) + a^2*b^4*e^(4*c))*e^(4*d*x) + 2*(a^6*e^(2*c) + 6*a^5
*b*e^(2*c) + 14*a^4*b^2*e^(2*c) + 16*a^3*b^3*e^(2*c) + 9*a^2*b^4*e^(2*c) + 2*a*b^5*e^(2*c))*e^(2*d*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)^3*(a + b/cosh(c + d*x)^2)^3),x)

[Out]

int(cosh(c + d*x)^6/(sinh(c + d*x)^3*(b + a*cosh(c + d*x)^2)^3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**3/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Integral(csch(c + d*x)**3/(a + b*sech(c + d*x)**2)**3, x)

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